Syllabus
* Markovian subdynamics
Slow versus fast degrees of freedom; derivation of the Smoluchowski equation for overdamped diffusion; thermal activation and noise-induced decay of metastable states - Arrhenius and Kramers theories; reduced dynamics of quantum subsystems - Davies weak-coupling theory, quantum Markov processes, rotating-wave approximation.
* Stochastic many-body models
Open interacting particle systems as mesoscopic models for diffusion; heat conduction in a chain of coupled oscillators with open ends; Kipnis-Marchior-Presutti model of transport.
* Autonomous macroscopic dynamics
Kinetic and diffusive (or "hydrodynamic") scaling; derivation of macroscopic diffusion equations from stochastic exclusion models; local thermodynamic equilibrium; equations of linear non-equilibrium thermodynamics.
* Macroscopic fluctuations
Generalized Onsager-Machlup theory of dynamical fluctuations; Onsager-Machlup symmetry breaking; non-equilibrium entropy from typical histories of large fluctuations, Hamilton-Jacobi equations for the entropy, current fluctuations, vortex patterns in two-dimensional transport.
Annotation
The second part is devoted to some aspects of the time-scale separation, which is responsible both for the
Markovian behavior of relevant ("slow") degrees of freedom for small open systems and for the autonomous dynamics of macroscopic systems. We also discuss basic stochastic models of interacting particles, their macroscopic limit and fluctuations.
For the 1st and 2nd year and for doctoral students.